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                <blockquote>
<p>本文主要介绍了数学建模美赛并分析了数模美赛历年的出题点</p>
</blockquote>
<p><img src="https://jack-1307599355.cos.ap-shanghai.myqcloud.com/img/image-20220112183134553.png" alt="2022美赛宣传海报" style="zoom:50%;"></p>
<h1 id="数学建模备赛：美赛基础入门及历年出题点分析"><a href="#数学建模备赛：美赛基础入门及历年出题点分析" class="headerlink" title="数学建模备赛：美赛基础入门及历年出题点分析"></a>数学建模备赛：美赛基础入门及历年出题点分析</h1><h1 id="一、美赛基础入门"><a href="#一、美赛基础入门" class="headerlink" title="一、美赛基础入门"></a>一、美赛基础入门</h1><h2 id="1-数模美赛入门"><a href="#1-数模美赛入门" class="headerlink" title="1. 数模美赛入门"></a>1. 数模美赛入门</h2><blockquote>
<p>对于零基础的人来说，还是先直观的认识一下数模美赛</p>
</blockquote>
<h3 id="A-2020-MCM-A-题目"><a href="#A-2020-MCM-A-题目" class="headerlink" title="A. 2020 MCM-A 题目"></a>A. 2020 MCM-A 题目</h3><p>在到点（比赛开始）之后，美赛官方会在官网上公布题目，即下面的PDF文件</p>
<p><img src="https://jack-1307599355.cos.ap-shanghai.myqcloud.com/img/image-20220112103650809.png" alt="赛题-1"></p>
<p><img src="https://jack-1307599355.cos.ap-shanghai.myqcloud.com/img/image-20220112103715739.png" alt="赛题-2"></p>
<p>英文看起来不太方便，所以翻译成中文来看看</p>
<blockquote>
<p><strong>2020MCM        A题：向北迁徙</strong></p>
<p>全球海洋温度影响某些海洋生物的生境质量。当温度变化太大，无法继续繁衍时， 这些物种就会迁移到其他更适合它们现在和未来生活和繁殖的生境。这方面的一个例 子是美国缅因州的龙虾种群,它们正在慢慢向北迁移到加拿大，那里较低的海洋温度提 供了更合适的栖息地。这种地理种群的变化会严重扰乱那些依赖生物稳定性的种群的 生计。</p>
<p>你的团队已被苏格兰北大西洋渔业管理财团聘请为顾问。该财团希望更好地了解， 如果全球海洋温度上升，苏格兰鲱鱼和鲭鱼可能从苏格兰附近的现有栖息地迁移的相 关问题。这两个鱼种对苏格兰渔业的经济贡献很大。鲱鱼和鲭鱼种群位置的变化，可 能会使以苏格兰为基地的小型渔业公司在经济上无法生存。因为这些公司使用的渔船 上没有冷藏设备，而他们需要收获新鲜鱼并将其运往苏格兰渔港的市场。</p>
<p>要求：</p>
<ol>
<li>建立一个数学模型，确定这两种鱼在未来50年内最可能迁徙到的位置，假设水温 会发生足够导致种群移动的变化。</li>
<li>根据海洋水温变化的速度，用你的模型预测最佳情况、最坏情况和最有可能的时 间：如果小型捕鱼公司继续在目前的地点作业，这些种群将太远，直到小型捕鱼公司 无法捕捞。</li>
</ol>
</blockquote>
<h3 id="B-最终解决方案"><a href="#B-最终解决方案" class="headerlink" title="B. 最终解决方案"></a>B. 最终解决方案</h3><p>经过四天的比赛，美赛要求参赛队伍最终提交一个PDF文件作为最终解决方案，其中包括Summary、论文、附录等等内容</p>
<p>下面是上面这道题的O奖文章之一</p>
<p><img src="https://jack-1307599355.cos.ap-shanghai.myqcloud.com/img/image-20220112104017921.png" alt="最终论文"></p>
<p>这是论文里的部分内容</p>
<p><img src="https://jack-1307599355.cos.ap-shanghai.myqcloud.com/img/image-20220112104359345.png" alt="部分论文内容" style="zoom: 67%;"></p>
<h3 id="C-美赛到底是什么？"><a href="#C-美赛到底是什么？" class="headerlink" title="C. 美赛到底是什么？"></a>C. 美赛到底是什么？</h3><p>在直观的认识了美赛之后，我们不禁想要问，什么是美赛？</p>
<blockquote>
<p><strong>美赛（美国大学生数学建模竞赛）是什么？</strong></p>
<ul>
<li>美赛=微型的科研项目</li>
<li>美赛=设计和策划方案</li>
<li>美赛=模型思路+程序+论文</li>
</ul>
</blockquote>
<p>下面我们就开始正式的介绍美赛</p>
<h2 id="2-美赛的发展及规模"><a href="#2-美赛的发展及规模" class="headerlink" title="2. 美赛的发展及规模"></a>2. 美赛的发展及规模</h2><p><strong>Mathematical Contest in Modeling（MCM）</strong>是一项由美国数学及其应用联合会(<strong>COMAP</strong>)发起组织的国际级竞赛项目，自1985年开始举办 ，每年一次。由美国自然基金协会和美国数学及其应用联合会共同主办 ，美国运筹学学会、工业与应用数学学会、数学学会等多家机构协办。 </p>
<p>1999年起，美国数学及其应用联合会又同时推出<strong>交叉学科竞赛（ Interdisciplinary Contest in Modeling ），简称 ICM</strong>（涉及数学、环境 科学、生物学、化学、资源管理等交叉学科）。</p>
<ul>
<li>MCM是国际大学生各类竞赛中级别最高的一项赛事，每年吸引世界 著名高校参赛，很多国内外著名科研机构和企业将其作为衡量学生素 质和水平的重要依据</li>
<li>美赛对国内大部分高校可以保研加分（虽然欲我无瓜<span class="github-emoji"><span>😂</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f602.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span>）</li>
<li>美赛成绩国外的认可度要高一些（出国留学）</li>
<li>国内高校对美赛的热情略低于国赛，大部分学校没有常规的的赛前集训</li>
</ul>
<p>美赛历年参加的人数在不断的增加：</p>
<ul>
<li>2015年，MCM近8000队，ICM近2000队</li>
<li>2016年，MCM近9000队，ICM近2000队</li>
<li>2017年，MCM 11000队，ICM6000多队</li>
<li>2018年，MCM 12000队，ICM7000多队</li>
<li>2019年，MCM14000多队，ICM11000多队</li>
<li><strong>人数逐年增加，选择MCM的人数多于ICM。</strong></li>
</ul>
<p>和国内一等奖最高不一样，美赛最高的奖项是特等奖。各个奖项以及获奖比利如下：</p>
<ul>
<li>特等奖（O奖）每题18、19队，&lt;1%</li>
<li>特等奖提名（F奖）每题180，234队，3%</li>
<li>一等奖（M奖） 5-6%，二等奖（H奖）22－26％</li>
<li>S奖（Successful Participant）</li>
</ul>
<p><strong>从2019年开始，严查违规问题（DQ），10%-20%被判违规，以前只有少量被判违规。</strong></p>
<h2 id="3-美赛的比赛形式及流程"><a href="#3-美赛的比赛形式及流程" class="headerlink" title="3. 美赛的比赛形式及流程"></a>3. 美赛的比赛形式及流程</h2><ul>
<li><strong>题目</strong>：各领域简化的应用问题<ul>
<li>MCM 3 题（A， B ，C，连续，离散，数据分析）</li>
<li>ICM 3 题（D，E，F，计算机网络，环境，政策措施）</li>
</ul>
</li>
<li><strong>提交成果</strong>：一篇包含<strong>问题分析、模型假设、建立、求解(数学软件)、 结果分析</strong>等的全英文论文。</li>
<li><strong>形式</strong>：3名大学生组队，在4天内完成通讯比赛</li>
<li><strong>标准</strong>：<ul>
<li>假设的合理性</li>
<li><strong>建模的创造性</strong></li>
<li>结果的正确性</li>
<li>表述的清晰性</li>
</ul>
</li>
</ul>
<p>2022年美赛的流程如下：</p>
<ul>
<li><strong>注册（报名）截至时间</strong>：比赛开始前两小时<ul>
<li><strong>2022年2月18日早上4点前</strong></li>
</ul>
</li>
<li><p><strong>比赛时间</strong>：一般美赛都在每年2-3月（过年前后，开学前）</p>
<ul>
<li><strong>2022年2月18日早上6点——2月22日早上9点</strong></li>
</ul>
</li>
<li><p>3月完成评审，4月30日之前COMAP网站发布结果</p>
</li>
<li>注册报名：<ul>
<li><strong>美赛官网注册报名</strong></li>
</ul>
</li>
</ul>
<p>COMAP官网：<a target="_blank" rel="noopener" href="https://www.comap.com/，第二个就是数模美赛">https://www.comap.com/，第二个就是数模美赛</a></p>
<p><img src="https://jack-1307599355.cos.ap-shanghai.myqcloud.com/img/image-20220112110317316.png" alt="COMAP官网以及美赛"></p>
<h2 id="4-美赛的能力要求与准备"><a href="#4-美赛的能力要求与准备" class="headerlink" title="4. 美赛的能力要求与准备"></a>4. 美赛的能力要求与准备</h2><p>首先，美赛对能力的要求如下：</p>
<ul>
<li>良好的数理知识（高等数学、概率论、线性代数）</li>
<li>编程及软件使用能力（Matlab、Python)</li>
<li>英语写作能力</li>
<li>资料搜集能力：一般除了C题其他的题目不会给数据，然而自己的模型是需要数据验证的，因此还要自己根据题目的暗示找数据</li>
<li>文献阅读能力</li>
</ul>
<p>接下来是美赛的准备方法：</p>
<ul>
<li>基础模型算法/编程</li>
<li>实战真题（3-5年）</li>
<li>阅读真题O奖论文：因为O奖论文是最接近评委心中的满分答案的论文</li>
</ul>
<p>而为了准备美赛，推荐的资料如下：</p>
<ul>
<li>数理知识：<ul>
<li>《数学建模算法与应用》司守奎等</li>
<li>历年美赛真题与O奖论文集合</li>
</ul>
</li>
<li>常用软件：<ul>
<li>Matlab, SPSS, Excel, PPT制图，Latex（必备）</li>
<li>Lingo，ArcGIS, Python，Origin, SAS, R语言（推荐）</li>
</ul>
</li>
<li>英语：<ul>
<li>雅思、托福写作词组+句式</li>
</ul>
</li>
</ul>
<h2 id="5-美赛的特点"><a href="#5-美赛的特点" class="headerlink" title="5. 美赛的特点"></a>5. 美赛的特点</h2><p>相对于国赛，美赛的特点如下：</p>
<ul>
<li>赛题灵活度高，内容广泛，开放性强</li>
<li><strong>对算法难度和准确度要求低于国赛</strong>，基本上合理即可</li>
<li><strong>注重方法的创新</strong></li>
<li>英文写作水平要求较高</li>
<li>注重论文的完整度与丰富度</li>
</ul>
<h2 id="6-美赛的组队与分工"><a href="#6-美赛的组队与分工" class="headerlink" title="6. 美赛的组队与分工"></a>6. 美赛的组队与分工</h2><p>大体来说，完成美赛需要团队具有下面三个模块的能力</p>
<p><img src="https://jack-1307599355.cos.ap-shanghai.myqcloud.com/img/image-20220112111608785.png" alt="美赛需要的三个能力"></p>
<p>在细节上，建议：</p>
<ul>
<li>至少一位擅长英语写作</li>
<li>至少一位擅长编程</li>
<li>建模知识都需要入门</li>
<li>根据你们的特长调整分工及选题</li>
<li><strong>最终决策权</strong></li>
<li>沟通、负责</li>
</ul>
<h1 id="二、美赛出题点分析"><a href="#二、美赛出题点分析" class="headerlink" title="二、美赛出题点分析"></a>二、美赛出题点分析</h1><p>本文只是对美赛的出题点进行了简要的介绍，跟多内容请参考其他文字</p>
<h2 id="1-美赛出题领域"><a href="#1-美赛出题领域" class="headerlink" title="1. 美赛出题领域"></a>1. 美赛出题领域</h2><p>从10年开始，历年美赛出题的领域如下：</p>
<ul>
<li><strong>工程</strong>：<ul>
<li>2018A多跳HF无限电传播</li>
<li>2016A浴缸的水温模型</li>
</ul>
</li>
<li><strong>交通</strong>：<ul>
<li>卢浮宫疏散</li>
<li>2017B高速收费站合并</li>
<li>2018D机场安检站优化</li>
</ul>
</li>
<li><strong>生态环境</strong>： <ul>
<li>2020A向北迁徙</li>
<li>2019A饲养一条龙</li>
<li>2019E环境恶化的代价 是什么？</li>
</ul>
</li>
<li><strong>金融</strong>：<ul>
<li>2016C优质基金挑战</li>
</ul>
</li>
<li><strong>新能源</strong>：<ul>
<li>2011C电动汽车</li>
</ul>
</li>
<li><strong>犯罪</strong>：<ul>
<li>毒品危机</li>
</ul>
</li>
<li><strong>疾病医学</strong>：<ul>
<li>2015A 根除埃博拉病毒</li>
</ul>
</li>
<li><strong>人文社科</strong>：<ul>
<li>2020F难民危机，2019B语言传播趋势, 2018F隐私成本问题</li>
</ul>
</li>
</ul>
<h2 id="2-美赛常见题型"><a href="#2-美赛常见题型" class="headerlink" title="2. 美赛常见题型"></a>2. 美赛常见题型</h2><p>其实不光是美赛，整个数学建模常见的题型如下：</p>
<ul>
<li><strong>运筹优化类问题</strong><ul>
<li>规划模型</li>
<li>排队论</li>
<li>神经网络</li>
<li>启发式算法</li>
</ul>
</li>
<li><strong>评价类问题</strong><ul>
<li>层次分析法</li>
<li>聚类分析法</li>
<li>主成分分析评价法</li>
</ul>
</li>
<li><strong>预测类问题</strong><ul>
<li>灰色预测</li>
<li>回归预测</li>
<li>神经网络</li>
<li>马尔科夫链</li>
</ul>
</li>
<li><strong>机理分析类问题</strong>：其实涵盖的面更广<ul>
<li>回归</li>
<li>拟合</li>
<li>微分方程</li>
</ul>
</li>
</ul>
<h2 id="3-美赛必备算法模型"><a href="#3-美赛必备算法模型" class="headerlink" title="3. 美赛必备算法模型"></a>3. 美赛必备算法模型</h2><p>为了应对美赛，每一小问都能划归到上面的四类题型。因此我们需要掌握下面的算法模型：</p>
<ol>
<li><strong>蒙特卡罗/计算机仿真算法</strong><ul>
<li>该算法又称随机性模拟算法，是通过计算机仿真来解决问题的算法， 同时可以通过模拟可以来检验自己模型的正确性，比较好用的算法</li>
</ul>
</li>
<li>数据拟合、参数估计、插值等数据处理算法<ul>
<li>比赛中通常会遇到大量的数据需要处理，而处理数据的关键就在于这 些算法，通常使用Matlab作为工具</li>
</ul>
</li>
<li>线性规划、整数规划、多元规划、二次规划等规划类问题<ul>
<li>建模竞赛大多数问题属于最优化问题，很多时候这些问题可以用数学 规划算法来描述，使用Matlab或Lingo软件实现</li>
</ul>
</li>
<li>网格算法和穷举法<ul>
<li>当重点讨论模型本身而轻视算法的时候，可以使用这种暴力方案，最 好使用一些高级语言作为编程工具</li>
</ul>
</li>
<li>最优化理论的三大非经典算法：模拟退火法、神经网络、遗传算 法<ul>
<li>这些问题是用来解决一些较困难的最优化问题的算法，对于有些 问题非常有帮助，但是算法的实现比较困难，需慎重使用</li>
</ul>
</li>
<li>排队论<ul>
<li>对算法要求高，如元胞自动机</li>
</ul>
</li>
<li>评价类算法<ol>
<li>层次分析法（AHP）（必掌握）<ul>
<li>作决策，去哪旅游，通过指标，综合考虑作决策</li>
</ul>
</li>
<li>主成分分析（必掌握）<ul>
<li>评价多个对象的水平并排序，指标间关联性很强</li>
</ul>
</li>
<li>其他<ul>
<li>熵权法（客观）、模糊综合评判（简单）、秩和比综合评价 法、TOPSIS法)、数据包络（DEA）、投影寻踪综合评价法、或者揉 和多种算法，比如遗传算法、最优化理论</li>
</ul>
</li>
</ol>
</li>
<li>预测算法<ol>
<li>灰色预测模型（灰箱模型）<ul>
<li>适合数据样本点个数少，数据呈现指数或曲线的形式</li>
</ul>
</li>
<li>回归分析预测（必掌握）<ul>
<li>求一个因变量与若干自变量之间的关系，若自变量变化后，求因变量 如何变化； 对样本点的个数、分布有要求</li>
</ul>
</li>
<li>其他<ul>
<li>微分方程预测、马尔科夫预测、时间序列预测、小波分析 预测、神经网络预测</li>
</ul>
</li>
</ol>
</li>
<li>多元分析与关联分析<ol>
<li>聚类分析（必掌握，参考19）</li>
<li>主成分分析（必掌握）</li>
<li>因子分析（必掌握）</li>
<li>典型相关分析</li>
<li>灰色关联分析法</li>
<li>方差分析、协方差分析</li>
<li>Person相关（样本点的个数比较多）</li>
</ol>
</li>
</ol>
<h2 id="4-美赛考察点"><a href="#4-美赛考察点" class="headerlink" title="4. 美赛考察点"></a>4. 美赛考察点</h2><p>美赛的考察点，或者说评判时候裁判看中的点，有下面几个：</p>
<ol>
<li>运筹优化、机理分析类问题（重难点）：相对于预测类问题，这两类问题能够处理的问题、包含的算法很广，因此是考察的重点</li>
<li>评价类问题常规要突出新意：因为现在评价模型已经非常完备了，因此使用评价类算法进行评价的时候要突出新意</li>
<li>模型的完整性和叙述的逻辑性：论文要好好写</li>
<li>数据分析、挖掘</li>
<li>方法迁移：把其他学科的研究方法、算法迁移过来，比如Word Embedding</li>
<li>创新：有理有据，打破常规</li>
<li>画图美观很重要</li>
<li>使用专业的软件</li>
</ol>

                
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                        <span class="card-title">数学建模备赛：数学建模题型划分、常用算法分类及其适用场景</span>
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                            本文主要讲解了数学建模的题型划分、常用算法分类以及常用算法的适用场景
                        
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